Reliable estimates in the anisotropic heat conduction problems
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The heat conduction problems for anisotropic bodies are studied taking into account the uncertainties in the material orientation. The best estimations of the upper and lower bounds of the considered energy dissipation functional are based on developing new approach consisting in solution of some optimization problems and finding the extremal internal material structures, which realize minimal and maximal dissipation. The motivation of this study comes from paper making processes, and more precisely, drying process, which consumes about 50% of the energy fed into the paper machine. The understanding of the effect of uncertainties in the process arises from structural properties of paper will provide the possibility to optimize the drying system.
KeywordsUncertainties Reliable estimates Heat conduction Anisotropic material
The problems of incompleteness of data and uncertainties are typical for anisotropic solids and structures having chaotic orientation of small material particles such as grains, crystal or short nanofibers. Different possible compositions of elementary particles with various orientations result in different values of such integral characteristics as a total dissipation energy in the heat conduction problems, total potential energy in the thermoelasticity and thermoconductivity problems. Taking into account the conditions of uncertainties concerning the material orientations it is very important to obtain various estimations of the considered functionals and in particular limiting estimates known as double-sided or bilateral estimates (see book by Banichuk and Neittaanmäki ).
The motivation of this study arises from understanding of paper making processes. As is well known that paper product have an anisotropic fibrous structure which properties depend on the making process and its parameters (velocity, tension, etc.). The understanding of heat conduction behaviour in anisotropic material is very critical for optimization of the system. During paper making, the drying process consumes about 50% of the energy fed into the paper machine; it is the single largest consumer of energy in the paper manufacturing process.
To model the drying of a moving paper web, several models exist in the literature (see e.g. Karlsson , Lampinen and Toivonen  and Lu and Shen ). For a thorough engineering-oriented discussion on paper drying, see the book edited by Karlsson . In our study, we have assumed, that the material is not moving. Moreover, we consider the fundamental mathematical setup of the problem, that the results can be applied widely.
In this article, the problem of estimation of dissipation energy characteristics is considered for anisotropic body constituting of the locally orthotropic material. It is assumed that an orientation of the principle axes of orthotropy is not known beforehand at each point of the body and can be distributed by various ways in different parts of the body including chaotic orientation. The search for double-sided estimates is reduced to the solution of optimization problems and finding the extremal orientations of the orthotropy axes.
Heat conduction problem for bodies from locally orthotropic material
Note the second boundary condition in (2) plays the role of transversality condition for the functional (4) and is satisfied ‘automatically’ for extremum solution. Note that Equation 3 is the Eulerian equation for the functional (4).
Uncertainties in orientation of orthotropic material and double-sided estimates
for any realization of Q satisfying the condition (7).
where min and max with respect to Q in Equations 16 and 17 are determined under constraint (7). Operation min with respect to φ in Equations 16 and 17 is performed taking into account boundary conditions from Equation 2.
and analyze extremum conditions and behavior equations.
Extremal conditions for orthotropic material orientation
and the symbol ⊗ is the tensor product.
where λ is some scalar value.
Double-sided estimates based on derived extremal conditions
corresponding to conditions (2) with the relations (32) constitute the conventional boundary value problem describing, as it is well known, homogeneous or nonhomogeneous isotropic processes of the heat conductivity. Under some known additional constraints superimposed on the boundary shape Γ=Γ g +Γ i , where Γ g ∩Γ i =0, we have the existence and uniqueness of the solution of (34) and (36) with given λ i .
for Laplace equation with mixed (in general case) boundary conditions. Here Δ is a Laplace operator acting in a three-dimensional space.
and for each separate subdomain Ω i , the same extremum way of material orientation is taken, then the isotropic heat conduction process is realized for all considered subdomains.
Two-dimensional case of extremal material orientation
corresponds to the larger eigenvalue λ2(λ2>λ1).
Examples of double-sided estimates
and r0 is a unit vector, oriented in radial direction.
and is an unit vector of the x3-axis, obtained when the vector x3 is divided by its length |x3|.
In the case, when the coefficient D ij and the considered eigenvalues λ i are independent of x=(x1, x2, x3), then the anisotropic behaviour equation is reduced to the Laplace equation which describes the heat conduction of homogeneous isotropic body. Since the theory of the heat conduction of isotropic homogeneous solids is well developed and solution of the corresponding boundary value problem has been found (analytically and numerically) for most problems of practical importance, then this reduction allows to consider the above problem of obtaining of double-sided estimates to be solved.
Taking into account the conditions of uncertainties concerning material orientations, we obtain various estimations of the considered functionals and in particular limiting estimates known as double-sided or bilateral estimates. The search of double-sided estimates as it was shown is reduced to the solution of optimization problems and finding the extremal orientation of the orthotropy axes. The results can be applied for example to the optimization of the paper drying process, which has a significant role in energy consumption of the paper production.
The research was performed under financial support of RFBR (grant 11-08-00030-a), RAS Program 12, Program of Support of Leading Scientific Schools (grant 2611.2012.1) and Funding from Academy of Finland (grants no. 140221 and 269351).
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